Infinite sample space in probability

Question

My text book on probability says "sample spaces with an infinite number of elements are quite common. As an example, consider throwing a dart on a square target and viewing the point of impact as the outcome."

Assuming 'point of impact' denotes the point in the square target where the dart hits, does this infiniteness pertain to infinite number of points in a square? Can anyone give any other example of

• Infinite number of elements in sample space, for a single well defined experiment (not like throwing a dice infinite number of times.)
• "When dealing with probabilistic models involving an uncountably infinite sample space, there are certain unusual subsets for which one cannot associate meaningful probabilities." Any example for this case?

Answer 1

Yes, the example given is making reference to the fact that there are an uncountable number of points inside a square on the real number plane.

A classic example of a sample space of infinite size is any continuous measurement, my favourite being human height. Theoretically, if your single experiment is measuring the height of the next person you meet, you could expect to see any measurement (of which there are an uncountable amount) between (let's be conservative here) 50 and 500 cm.

And I think the second point is hinting at the fact that not every subset of the real numbers is measurable. (See Vitali Set - Wikipedia for an example.) This is quite advanced stuff, but basically the foundation of probability is measure theory, which is the mathematical tool for measuring the size of a subset. Unfortunately, for the kind of measurement we usually use on subsets of the real numbers, (in short, for an interval $(a,b)$ its size is $b-a$) one can construct a subset of the reals like the Vitali set for which there is no sensible notion of its size.